The lifetime distribution

The Relationship between the Lifetime Distribution and the Diffusion Equation 

An isolated ion-pair, of initial separation r0 at time t0 = 0, in a nonpolar solvent may recombine or separate and ultimately escape. At a time t, the probability that the ion-pair will have recombined is (t r0, t0 = 0) and that it is still extant p(t r0) f0 = 0). A short while later, the probability that the ion-pair has not recombined is p t + df r0, t0 =0). The change in survival probability is the probability that the ion-pair recombined during the time interval t to t + df, that it had a lifetime between t and t -f df. Defining the lifetime distribution function as f(f), then 

As time proceeds, the survival probability decreases and asymptotically tends to a constant value p(t- oo] r0, f0 = 0) =, the ultimate survival 

This is not unity because there is a probability, that the ion-pair will escape and so never recombine. 

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Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

By means of Laplace transforms of the foregoing three equations mating use of the convolution theorem and the assumptions Pf(t) — Pt a constant which is the ratio of the in use time (t the total operating time of the 4th component), Gt(t) si — exp ( — t/dj (note that a double transform is applied to Ff(t,x)), we obtain an expression in terms of the lifetime distribution, i.e.,... 

Fig. 3.11. FRET FLIM experiment to study colocalization of two lipid raft markers, GPI-GFP and CTB-Alexa594. The rows of images show intensity and lifetime images of donor-labeled and donor + acceptor-labeled cells. The histogram shows the lifetime distribution of the whole cells. The FRET...

A closer look at the data shows the lifetime distributions are comparatively broad, about 0.25 ns for both distributions. This is in fact much broader than what one would expect from photon statistics alone. Based on realistic / -values (1.2-1.5) lifetime images recorded with this many counts are expected to yield distributions with widths on the order of 0.1 ns. The broadening is therefore not because of photon statistics. Variations in the microenvironment of the GFP are the most likely source of the lifetime heterogeneities. Importantly, such sensitivity for local microenvironment may be the source of apparent FRET signals. In this particular FRET-FLIM experiment, we found that the presence of CTB itself without the acceptor dye already introduced a noticeable shift of the donor lifetime. Therefore, in this experiment the donor-only lifetime image was recorded after unlabeled CTB was added to the cells. The low FRET efficiency and broadened lifetime distribution call for careful control experiments and repeatability checks. 

Rzad et al.( 1970) compared the consequences of the lifetime distribution obtained by ILT method (Eq. 7.27) with the experiment of Thomas et al. (1968) for the decay of biphenylide ion (10-800 ns) after a 10-ns pulse-irradiation of 0.1 M biphenyl solution of cyclohexane. It was necessary to correct for the finite pulse width also, a factor rwas introduced to account for the increase of lifetime on converting the electron to a negative ion. Taking r = 17 and Gfi = 0.12 in consistence with free-ion yield measurement, they obtained rather good agreement between calculated and experimental results. The agreement actually depends on A /r, rather than separately on A or r. 

A solution of a pure fluorophore may reasonably be expected to display a single exponential decay time. The emission from fluorophore-protein conjugates, on the other hand, may be best characterized by two or three exponential decay times (Table 14.2). In labeling proteins with fluorophores, a heterogeneity of labeled sites results in fluorophore populations that have different environments, and hence different lifetimes. The lifetime distribution of a fluorophore-protein conjugate in bulk solution may vary further when immobilized on a solid support (Table 14.2). 

One would expect that lowering the temperature or increasing the viscosity of the solvent would increase the width of the lifetime distribution, since both factors may affect the rate of transitions between microstates. If this rate is high as compared with the mean value of the fluorescence lifetime, the distribution should be very narrow, as for tryptophan in solution. When the rate of transitions between microstates is low, a wide distribution would be expected. 

The measurement of fluorescence lifetimes is an integral part of the anisotropy, energy transfer, and quenching experiment. Also, the fluorescence lifetime provides potentially useful information on the fluorophore environment and therefore provides useful information on membrane properties. An example is the investigation of lateral phase separations. Recently, interest in the fluorescence lifetime itself has increased due to the introduction of the lifetime distribution model as an alternative to the discrete multiexponential approach which has been prevalent in the past. 

The meanings of Gn and Ggi are the number of the ion-pairs per 100 eV of radiation absorbed which never recombine together and the number of ion-pairs which recombine in the absence of a scavenger (or electron or cation) (see Sect. 3.3 and Fig. 32, p. 185). An expression of the form of eqn. (174) can be used to estimate the lifetime distribution of ion-pairs [365] or, rather more fundamentally, the initial distribution of ion-pair distances (see Sect. 3.3). Clearly, for small scavenger concentrations, G(P) is represented equally well by the two equations above. Comparing either with the probability of scavenging [eqn. (172)], it follows that... 

Equation (345) shows that the average scavenging probability of an ion-pair initially separated by a distance r0 is the Laplace transform (into k%c space) of the lifetime distribution function. 

To show how the lifetime distribution of eqn. (346) is directly related to the diffusion equation analysis, first substitute eqn. (343) into eqn. (346) and take the Laplace Transform... 

Here A is a constant from the normalization condition j a(r) = 1, rc is the center value of the lifetime distribution, and W is the full-width-at-half maximum for the Lorentzian function. 

Even though the lifetime distributions appear to be quite different, the recreated data are almost identical except for a small deviation near 200 ns and less obvious ones at shorter times. If the relative differences are plotted, systematic differences beyond the statistical noise are noticeable up to 200 ns, particularly when several channels are binned together. Given sufficient statistics, in principle, one can tell the difference between a bimodal and a monomodal distribution. The shown simulated spectra are based on 108 counts, five to 10 times the amount collected for the data discussed here. 

In practice the lifetime distributions are usually obtained using a computer program such as the MELT [21] or CONTIN [22, 23] programs. The reliablity of these programs for measurring the o-PS lifetime distribution in polymers was shown by Cao et al [24]. A detailed description of these methods of data analysis is presented in Chapter 4. The advantage of the continuous lifetime analysis is that one can obtain free volume hole distributions rather that the average values obtained in the finite analysis. 

We will consider the MWD in two simple cases. The first is when chain transfer is sufficiently rapid to ensure that all other chain-stopping events can be ignored. In such a situation, whereas the compartmentalized nature of the reaction may affect the rate of initiation of new chains, it will not affect the lifetime distributions of the chains once they are formed. The MWD may then be found from the bulk formulas, provided only that the average number of free radicals per particle, is known. Such an approach has been used by Friis et al. (1974) to calculate the MWD evolved in a vinyl acetate emukion polymerization. These authors included in addition the mechanisms of terminal bond polymerization and of transfer to polymer (both of which cause broadening). The formulas required for the in corporation of these mechanisms could be taken from bulk theory. 

At this point it is worthwhile to review the possible failures of RRKM theory [9, 14] within a classical framework. First, the dynamics in some regions of phase space may not be ergodic. In this instance, which has been termed intrinsic non-RRKM behaviour [38], the use of the statistical distribution in Eq. (2.2) is inappropriate. In the extreme case of two disconnected regions of space, with one region nonreactive, the lifetime distribution is still random with an exponential decay of population to a non-zero value. However, the averaging of the flux must then be restricted to the reactive part of the phase space, and the rate coefficient is then increased by a factor equal to the reciprocal of the proportion of the phase space that is reactive. 

A RRKM unimolecular system obeys the ergodic principle of statistical mechanics [337]. A quantity of more utility than N t), for analyzing the classical dynamics of a micro-canonical ensemble, is the lifetime distribution Pc t), which is defined by... 

This model is clearly incomplete, since it does not account for vague tori [355] and the complex Arnold web [357, 358] structure of a multidimensional phase space with both chaotic and quasi-periodic trajectories. However, Eq. (74) does properly describe that, with non-ergodic dynamics, the lifetime distribution will have an initial component that decays faster than the RRKM prediction as found in the simulations by Bunker [323,324] and the more recent study of HCO dissociation [51]. Additionally, there will be a component to the classical rate, which is slower than /srrkm, for example, in the dissociations of NO2 and O3 this component cannot be described by an expression as simple as the one in Eq. (74). 

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